Vertical distribution of solid flux in a snow-wind flow

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Vertical distribution of solid flux in a snow-wind flow Dyunin, A. K.

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S n o w d r i f t s can be a major s o u r c e of d i s r u p t i o n i n t h e opera.tl.on of t r a n s p o r t a t i o n s e r v i c e s and a g e n e r a l n u i s a ~ z c e i n t h e ' normal w i n t e r t i m e a c t i v i t y of a community. Such d r i f t s a r e formed whenever a wind, s t r o n g en.oug2.1 t o t r a n s p o r t h o r i z o n t a l l y a s i g - n i f i c a n t amount of snow, e n c o u n t e r s a n o b s t a c l e

which f o r c e s it t o d e p o s i t some of t h i s ' snow. The u s u a l approach talcen i n d e f e n d i n g a n a r e a o r s t r u c - t u r e a.gain.st; s n o w d r i f t i n g has been t o l o c a t e t h e s t r a u c t u r e p r o p e r l y s o t h a t t h e d r i f t problem w i l l be a minimum and t o e r e c t o b s t a c l e s , s u c h as snow- f e n c e s , t o c o r l t r o l where t h e snow w i l l be d e p o s i t e d . The approach t a k e n i n t h e development of t h e s e d e -

f e n c e s h a s been l a r g e l y e m p i r i c a l . ~ t t e n t i o n ' h a s been d i r e c t e d p r i m a r i l y t o t h e c h a r a c t e r of t h e a i r flow w i t h l i t t l e a t t e n t l o n b e i n g g i v e n t o t h e ma- t e r i a l t r a n s p o r t e d . I n some , c i r c u m s t a n c e s , i t would be a n a d v a n t a g e t o have a more complete d e f e n c e a - g a i n s t s n o w d r i f t i n g t h a n i s now a v a i l a b l e . I n t h e i r a t t e m p t s t o d e v e l o p t h i s d e f e n c e , e n g i n e e r s a r e g i v - i n g more c o n s i d e r a t i o n t o t h e t h e o r e t i c a l a s p e c t s of t h e problem and i n p a r t i c u l a r t o t h e r e l a t i o n - s h i p s between t h e a i r flow and t h e snow b e i n g t r a n s - p o r t e d

.

I t i s one of t h e r e s p o n s i b i l i t i e s of t h e Snow and I c e S e c t i o n o f t h e D i v i s i o n of B u i l d i n g R e s e a r c h t o c o l l e c t and make a v a i l a b l e i n f o r m a t i o n r e q u i r e d f o r t h e s o l u t i o n of snow and i c e problems e n c o u n t e r - ed i n e n g i n e e r i n g p r a c t i c e . T h i s r e p o r t g i v e s t h e - o r e t i c a l c a l c u l a t i o n s and e x p e r i m e n t a l o b s e r v a t i o n s on t h e t r a n s p o r t of snow by wind. S t p r e s e n t s f o r - mulae g i v i n g t h e dependence on wind speed and h e i g h t above ground of t h e amount of' snow moved i n a g i v e n t i m e . T h i s i s t h e second Iiussian p a p e r on t h i s sub-

J e c t t h a t h a s been p u b l i s h e d i n t h e N a t i o n a l Research Council t r a n s l a t i o n s e r i e s . The f i r s t p a p e r , a l s o by A . K . Dyunin, i s e n t i t l e d "Fundamentals of t h e Theory of Snowdraifting" (NRC T e c h n i c a l T r a n s l a t i o n

952)

.

The paper was t r a n s l a t e d by M r . G . Belkov of t h e T r a n s l - a t i o n S e c t i o n of t h e N a t i o n a l R e s e a r c h C o u n c i l L i b r a r y , t o whom t h e D i v i s i o n of B u i l d i n g Research w i s h e s t o r c c o r d i t s t h a n k s .

Ottawa,

Title:

Author:

Reference :

Translator:

NATIONAL FU33EAHCH COUNCIL OF CANADA

Technical Translation

999

Vertical distribution of solid flux In a snow-wind flow

(0 raspredelenii raskhoda snegovetrovogo potoka po vysote)

A.K. Dyunin

Trudy Transportno-Energeticheskogo Instltuta,

(4):

49-58, 1954

VERllICAL DISTRIHUTION 01' SOLID ITLUX I N A SNOW-WIND FLOW I n t h i s p a p e r a n a t t e m p t i s made t o a n a l y z e t h e o r e t i c a l l y t h e d i s t r i b u t i o n w i t h h e i g h t of t h e s o l i d f l u x i n a snow-wind f l o w . The a n a l y t i c a l r e s u l t s a r e compared w i t h f i e l d o b s e r v a t i o n s . The g e n e r a l e x p r e s s i o n f o r t h e s o l i d f l u x of a snow-wind f l o w q* a t a c e r t a i n h e i g h t above t h e s u r f a c e y h a s t h e form

where p i s t h e weight of snow c o n t a i n e d i n a u n i t volume of a i r Y a t h e i g h t y , i n g/m

,

7 i s t h e a v e r a g e t r a n s l a t i o n a l v e l o c i t y of snow p a r t i c l e s

Y

i n m/sec. L e t u s d e f i n e t h e v a l u e of p

.

Y

It h a s been e x p e r i m e n t a l l y proven t h a t r i v e r s i l t and a l s o s a n d , d u s t and snow i n moving a i r

(1'3'6)

a r e r a i s e d i n t h e form

I1

of c l o u d s " . The " c l o u d " of p a r t i c l e s i s c a s t up by s u r f a c e e d d i e s and p e n e t r a t e s i n t o t h e mean f l o w .

The d i m e n s i o n s 1 of t h e s e p e n e t r a t i n g masses i n c r e a s e i n a p - p r o x i m a t e p r o p o r t i o n t o t h e h e i g h t y which i s confirmed by t h e ob- s e r v a t i o n s of P r o f . A . Y a . M i l o v i c h ( I , page

33),

i . e . we have t h e f o l l o w i n g dependence where a and f3 a r e c o n s t a n t p a r a m e t e r s . Assume t h e p e n e t r a t i n g rnasses ( " c l o u d s " ) t o be c y l i n d e r s w i t h a c r o s s - s e c t i o n a l a r e a of a 12 ( s e e F i g . 1) and c o n s i d e r a p l a n e

*

I n r a i l t r a n s ~ o r t l i t e r a t u r e t h e s o l i d f l u x of a snow-wind f l o w i s f r e q u e n t l y - c a l l e d " i n t e n s l t y of snow t r a n s f e r " and i s meas- u r e d i n g/crn ml.n.

s e c t i o n .

Let p be t h e weight of snow p a r t i c l e s i n a u n i t of volume of t h e p e n e t r a t i n g mass ( a t t h e h e i g h t of i t s c e n t r e of g r a v i t y y ) .

Then t h e t o t a l welght of p a r t i c l e s p e r u n i t of c y l i n d e r l e n g t h

When t h e h e i g h t i s i n c r e a s e d by Oy, we have

N e g l e c t i n g t h e s m a l l components of h i g h e r o r d e r we g e t

F),+dy

=

a p y f 3 - 1 - 2 a I p y - A / T a 1 2 . Ap.

Assuming t h a t t h e f l u i d ( a i r ) o u t s i d e t h e p e n e t r a t i n g cloud has a n e g l i g i b l e q u a n t i t y of i m p u r i t i e s , we f i n d t h a t t h e t o t a l weight of p a r t i c l e s i n t h e "cloud" does n o t change, i . e .

~ u - p y + ~ ,

from which one can immediately o b t a i n

2a lp, . A I -- - a i s . A p .

P a s s i n g t o t h e l i m i t when O ~ - * O , we g e t t h e d i f f e r e n t i a l e q u a t i o n

But a c c o r d i n g t o t h e f o r e g o i n g 1 = ay f B and d l = ady,. Consequent-

I n t e g r a t i n g we g e t

Let = c , where c i s a l i n e a r magnitude. To d e f i n e t h e a r b i - t r a r y c o n s t a n t co we s t a t e t h e boundary c o n d i t i o n s ; Py = Po when y = O . Then

This forainula was derlived by V .N. Goncharov (reference (1)

,

page 194). Hcrc 1% is given In

a

:;ornewhat altered and simplified f orrn

.

The same author also derived the following phenomenalogical formula for determining v (reference (l), page 160).

Y

Let a particle move at the velocity of v at the height y

Y

where the rate of flow is v

.

The aerodynamic forces F acting on Y'

the particle, as was shown theoretically by S.A. Chaplygin

( 5 )

and experimentally confirmed by A.I. Losievskii ( 2 )

,

depend on the square

2

of the relative velocity (V

-

_vy)

.

Y'

where a , is a constant coefficient for the given particles and fluid

.

If the mass of particles is equal to m, then according to the impulse theory ~ d t . - - m.dv,,, Expressing F by

(3)

we get - . After integration 1

+'?

n, S \ I y - v '- -Y' I+-!'!! a, c

where s = v t, c' is an integration constant. We assume that

Y'

during the period of time t there will be no change in the flow direction. In a quasi steady state turbulent flow the averaged magnitude

s

=

v

t can be considered constant.

Y'

Then

vy = T "YI

where c'rn/a,.s

I --I-

-

For t h e d c f l n i t i o n of' v w e u s e t h e logarithmic formula of P r o f .

Y

S . A . Sapozhnllcova ( r e f e r e n c e ( 4 ) , page 7 2 ) :

( 5 )

where

v,

-

t h e average v e l o c i t y a t u n i t h e i g h t above t h e ground ( f o r example 1 m e t r e ) ,

6

-

a l i n e a r c h a r a c t e r i s t i c of t h e roughness of t h e s u r f a c e . Consequently

One should keep i n mind t h a t t h i s formula i s v a l i d o n l y f o r v a l u e s of y > 6 . A schematic graph of t h e f u n c t i o n ( 6 ) i s shown i n F i g . 2 .

A t a c e r t a i n h e i g h t y o . t h e s o l i d f l u x of a flow r e a c h e s a maximwn qo and t h e n d e c r e a s e s a s y m p t o t i c a l l y approaching t h e Oy a x i s . The magnitude yo we f i n d by t a k i n g t h e d e r i v a t i v e w i t h r e - Ig f? ( y o

-+

c)---- -- 2 (Ig y - Ig 2 , ) , r) Y 0 Whence :

The graph shown i n F i g .

3

was constr'ucted a c c o r d i n g t o t h i s f o r m u l a . With t h i s graph, knowing c and 6 , we can d e t e r m i n e y o .

The v a l u e of a maximwn f l u x qo i s

D i v i d i n g

( 6 )

by

( 8 )

w e g e t

k'or d e t e ~ m i n i n g t h e t o t a l s o l l d discharge Q i n a wind-snow

Y

stratum froin 6y to y one must Integrate equation

(9)

within this range :

Keeping In mind

(7)

one can write

when y -+

I n I f -

( ;)

,,-,,,

.Qm = 240 Yo (yo -1- c) 7---

We will show that for the purpose of simplifying calculations in processing the experimental data one can replace formula ( 9 ) with the following approximate formula which is structurally simi- lar to expression (2):

where

Formula (12) is valid only under the condition yo< y. Let us assume that in the range of O < y < yo the flux q depends linearly on height y,

Y

Y = q o - Y o

Substituting

(13) in

( 1 5 ) wc g e t (11).

F i g . 4 shows a comparison of g r a p h s p l o t t e d from formulae

( 9 )

and ( 1 2 ) u s i n g v a r i o u s v a l u e s of c and 6 . Curves c o r r e s p o n d i n g t o formula ( 1 2 ) a r e p l o t t e d i n dashed l i n e s . I t can be r e a d i l y s e e n t h a t t h e dashed l i n e c u r v e s a r e v e r y c l o s e t o t h e s o l i d l i n e c u r v e s c o r r e s p o n d i n g t o formula ( 9 ) . The v a l u e of c , when c and 6 a r e known can be d e t e r m i n e d from t h e g r a p h ( F i g .

5 )

p l o t t e d from e x p r e s s i o n

(13)

and ( 7 ) .

The e x p e r i m e n t a l d a t a of TsNII MPS ( c e n t r a l Research I n s t i t u t e of t h e M i n i s t r y of ail roads) and TEI ZSFAN ( T r a n s p o r t a t i o n - p o w e r I n s t i t u t e of t h e E a s t S i b e r i a n S e c t t o n o f t h e Academy of s c i e n c e s ) make i t p o s s i b l e t o f i n d n u m e r i c a l l y t h e a v e r a g e v a l u e of c and 6

f o r a snow-wind f l o w .

F i g .

6

shows t h e d a t a o f anemometer measurements o f wind ve- l o c i t i e s a t v a r i o u s h e i g h t s . The c i r c l e s w i t h d o t s a r e p l o t t e d from t h e d a t a of TsNII MPS ( o r e n b u r g Railway, F e b r u a r y 1952) and t h e s o l i d c i r c l e s from t h e d a t a of T E I ZSFAN ( ~ o m s k Railway, Decem- b e r 1952

-

J a n u a r y

1953).

I n e a c h c a s e t h e v e l o c i t y a t t h e h e i g h t of 0 . 5 m i s t a k e n as u n i t y . The a b s o l u t e v a l u e s of t h e s e v e l o c i t i e s v a r y w i t h i n t h e r a n g e of 4 . 4

-

1 0 . 7 m/sec. F i g .

6

shows t h a t t h e wind v e l o c i t y o v e r a smooth snow s u r f a c e h a s on t h e a v e r a g e a l o g - a r i t h m i c dependence on t h e h e i g h t y .

S u b s t i t u t i n g i n formula

( 5 )

t h e dependence of wind speed on h e i g h t g i v e n i n F i g .

6,

we f i n d t h a t l o g 6 =-4.85, 6 = 0.00141 cm.

B l i z z a r d d a t a w a s used t o d e t e r m i n e c . The b l i z z a r d m e t r e of t h e t y p e VO, used by TsNII MPS* and TEI ZSFAN, h a s a n e n t r a n c e a p e r t u r e 2 cm h i g h . By p l a c i n g t h e n o z z l e r i g h t on t h e snow s u r - f a c e ( " s u r f a c e b l i z z a r d m e t r e " ) we do n o t c o n s e q u e n t l y d e t e r m i n e t h e maximum f l u x qo b u t t h e a v e r a g e f l u x qO-*

,

which i s d e t e r m i n e d t h e o r e t i c a l l y from formula ( 1 0 )

wllerc y o , c

,

6 a r c expre:;scti Ln c e n t l r n c t r e s

.

V . N . Conctlnrov found ( r e f c r c n c e ( l ) , page 4 1 ) t h a t t h e v a l u e of c i n a 1iqut.d flow depends primarily on t h e d e p t h of t h e l i q u i d .

P I G . 7 shows t h e d a t a o f V . N . Goncharov o b t a i n e d by two d l f i ' e r e n t methods. T h e more r e l i a b l e of t h e methods was t h a t of suspendcd p a r t i c l e s , t h e a n a l y s i s o f which w i l l n o t be d e a l t w i t h h e r e . * The maximum d e p t h of f l o w i n t h e experiments of V . N . Goncharov was

44

cm. For t h i s d e p t h , c = 2 . 5 cm.

P i g . 7 a l s o c o n s i d e r s t h e dependence o f c on t h e r o u g h n e s s of t h e s u r a f a c e . It i s interesting t o n o t e t h a t f o r a snow-wind f l o w , t h e c a l c u l a t i o n s g i v e c

-

2 . 7 cm on t h e a v e r a g e , a b o u t t h e same v a l u e a s f o r flow i n a v e r y deep f l u i d body.

I n T a b l e I t h e t h e o r e t i c a l l y c a l c u l a t e d v a l u e s of

3

a r e qo-2

compared w i t h t h e c o r r e s p o n d i n g mean v a l u e s of t h i s r a t i o c a l c u l a t e d from experimental. d a t a of TsNII MPS and TEI ZSFAN. E'or t h e t h e o r e t i - c a l c a l c u l a t i o n i.t was assumed t h a t c = 2 . 8 cm; 6 = 0.001.41 cm. Ac-

N

c o r d i n g t o t h e g r a p h i n P i g . j yo ; 0 . 2 8 crn. The c o r r e s p o n d i n g v a l u e

of c , =

4 . 7

cm.

According t o formu1.a ( 1 6 ) qo-* = 0 .767 qo

.

These d a t a a r e shown i n F i g .

8.

The r e s u l t s of o b s e r v a t i o n s a g r e e s a t i s i ' a c t o r i . l y w i t h t h e o r y .

The v a l u e of c a p p a r e n t l y d o e s n o t depend i n any e s s e n t i a l way on t h e a b s o l u t e v a l u e of t h e v e l o c i t y of' tkie snow-wind flow as can be s e e n from F i g .

9

where t h e t h e o r e t i c a l c u r v e s q = f ( Y , v , ) a r e compared wi.l;h e x p e r i m e n t a l l y found p o i n t s of s i n g l e m e a s u r e n ~ e n t s of a s o l i d f l u x a t v e l o c i t l c s oT' v, ( a t t h e h e i g h t o f 1 m ) 6 . 3

-

12 m/

s e c . The b l i z z a r d measurements wcrc c a r r l e d o u t by T E I ZSFAN.

I.. The d i s t r l i b u t i o n wl.t'L-1 h e i g h t or' t h e s o l i d f l u x of' a snow-

----

---

*

The v a l u e of' c

,

as shown by V

.N

.

Goncharov, c h a r a c t e r l z c s t h e chnngc w i t h h e i g h t 01' a

:I

lllcar v e r t i c a l . p u l s a t i n g component of tkie v e l o c i t y o r turbul.cnt; ['A ow

.

wind flow, as any other mixed flow, depends on the profile of the averaged translational vel-ocities and the linear characteristic of turbulence C of the wind flow.

2. The precise expression for the total solid flux in a layer

0

-

y has the form

This formula can be replaced by the approximate formula

3.

Curves constructed from these formulae agree satisfactorily with experimental data obtained under field conditions.

4 . In using these formulae one can carry out 'he maJor part of the blizzard-measuring observations with the use of surface-bliz- zard measuring instruments and from these data one can precisely determine the total solid flux of a snow-wind flow.

References

---

1. Goncharov, V.N

.

Dvizhenie nanosov ( ~ h e movement of silt), 1938. 2. Losievskii, A.l. Laboratornoe issledovanie protsessov obrazo-

vaniya perekatov (~aboratory lrivestigation of the formation of sand bars), 1934.

3.

Prandtll

,

I,. Gidroaerbornckhanika, Chapter

5,

1951.

4.

SapozhnLkova, S . A . Mikrokllmat i mestnyi klimat (~icroclimate and local climate), 1950.

5.

Chaplygin, S . A .

K

teorli metelei (on the theory of blizzards). Sobr. such. Vol. 11,

1948.

Table I

The motion of penetrating masses saturated with admixtures

Organizations carrying out observations TsNII MPS TEI Z S F A N I I TsNII MPS TEI ZSFAN I I I I I I Y J cm 0 - 2

6

6

11 16 21 31 41 51 No. of observations

43

6

47

43

10

4

4

4

x.

in0 (10-2 Mean of experimental data 100

26

-6

24.2 11.55

5

-54

4

5'7 2.1 0.49

0.45

Theoretical calc

.

100 25.2

25.2

10

.g

6.1

4

.O 2

.o

1.2 0.8

Fig. 2

A

schematic graph of the function q = f(y)

P l g . 3

F i g .

4

A g r a p h comparing f o r m u l a e

( 9 )

and ( 1 2 )

F i g .

5

A p r o f i l e of average t ; r ' a t ~ u L a t i o n a L v c : l o c i t i c s of' t h e snow-wj nd f l o w a c c o r d i n g to t h e du1,a of 'LIE_I %SFAN a n d TsNLI. MPS

Graph C = f (h)

0 o The value of C determined by the method of measuring the falling rate of particles

0

0 _{o o} in

a

flow.

(on

a smooth surface and on a

a m e V

+

surface made up of 5

-

r/ rnrn grain sizes)

v

+

The value of'

C

determined by the method of suspended particles

e

A

smooth surface and a surface of

5

-

7

rnrn

grain sizes

v Surface of 18

-

12 mrn grain sizes

+

Surface of

31

-

22 mrn grain sizes

1

o The average value of C for a snow-wind flow

Pig. '7

Value of c depending on tile depth of flow (according to

the

data of

V

.N. Goncharov)

Fig.

8

f * CO

"1

Graph of t h e f u n c t i o n -LL- 100 = f ( y ) qo- 2 m . . o V, = 1 2 m/sec

----

x V,

= 11.28 rn/sec

-

.-.

V, =

8.73

m/sec

....

V, =

6 . 3

m/sec F i g .

9

Graph of' t h e I'unct Lon q =: f ' ( y - v , )

I I -- The t h e o r e t i c a l curve x = 2 . 8 cm x Experimental d a t a of T E I ZSPAN

=

Experimental d a t a of T s N I I MPS I5 - -

k

t----

n sg r J t . f m : 5'04